Abstract:This talk will consider the delay differential equation (DDE) ˙ x(t) = −g(x(t))+f (x(t − τ )) which share the same equilibria with the corresponding ordinary differential equation (ODE) ˙ x(t) = −g(x(t)) + f (x(t)). For the bistable case, both the DDE and ODE share three equilibria x=0 < x1< x2 with x =0 and x2 being stable and x1 being unstable for the ODE. We are concerned with stability of these equilibria for the DDE and the basins of attraction of x0 and x2 when they are asymptotically stable for the DDE. Combining the idea of relating the dynamics of a map to the dynamics of a DDE and invariance arguments for the solution semiflow, we are able characterize some subsets of basins of attraction of these equilibria for the DDE. In addition, existences of heteroclinic orbits are also explored. The general results are applied to a particular model equation describing the matured population of some species demonstrating the Alee effect.